Algebra Examples

$f (x) = - \frac{1}{2} \sqrt{x + 3}$

Step 1

Write $f (x) = - \frac{1}{2} \sqrt{x + 3}$ as an equation.

$y = - \frac{1}{2} \sqrt{x + 3}$

Step 2

Interchange the variables.

$x = - \frac{1}{2} \cdot \sqrt{y + 3}$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \frac{1}{2} \cdot \sqrt{y + 3} = x$ .

$- \frac{1}{2} \cdot \sqrt{y + 3} = x$

Step 3.2

Multiply both sides of the equation by $- 2$ .

$- 2 (- \frac{1}{2} \cdot \sqrt{y + 3}) = - 2 x$

Step 3.3

Simplify the left side.

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Step 3.3.1

Simplify $- 2 (- \frac{1}{2} \cdot \sqrt{y + 3})$ .

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Step 3.3.1.1

Combine $\sqrt{y + 3}$ and $\frac{1}{2}$ .

$- 2 (- \frac{\sqrt{y + 3}}{2}) = - 2 x$

Step 3.3.1.2

Cancel the common factor of $2$ .

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Step 3.3.1.2.1

Move the leading negative in $- \frac{\sqrt{y + 3}}{2}$ into the numerator.

$- 2 \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 3.3.1.2.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 3.3.1.2.3

Cancel the common factor.

$2 \cdot - 1 \frac{- \sqrt{y + 3}}{2} = - 2 x$

Step 3.3.1.2.4

Rewrite the expression.

$- - \sqrt{y + 3} = - 2 x$

Step 3.3.1.3

Multiply.

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Step 3.3.1.3.1

Multiply $- 1$ by $- 1$ .

$1 \sqrt{y + 3} = - 2 x$

Step 3.3.1.3.2

Multiply $\sqrt{y + 3}$ by $1$ .

$\sqrt{y + 3} = - 2 x$

Step 3.4

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{y + 3}}^{2} = {(- 2 x)}^{2}$

Step 3.5

Simplify each side of the equation.

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Step 3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y + 3}$ as ${(y + 3)}^{\frac{1}{2}}$ .

${({(y + 3)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Step 3.5.2

Simplify the left side.

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Step 3.5.2.1

Simplify ${({(y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.1.1

Multiply the exponents in ${({(y + 3)}^{\frac{1}{2}})}^{2}$ .

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Step 3.5.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(y + 3)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 3.5.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.5.2.1.1.2.1

Cancel the common factor.

${(y + 3)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Step 3.5.2.1.1.2.2

Rewrite the expression.

${(y + 3)}^{1} = {(- 2 x)}^{2}$

Step 3.5.2.1.2

Simplify.

$y + 3 = {(- 2 x)}^{2}$

Step 3.5.3

Simplify the right side.

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Step 3.5.3.1

Simplify ${(- 2 x)}^{2}$ .

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Step 3.5.3.1.1

Apply the product rule to $- 2 x$ .

$y + 3 = {(- 2)}^{2} x^{2}$

Step 3.5.3.1.2

Raise $- 2$ to the power of $2$ .

$y + 3 = 4 x^{2}$

Step 3.6

Subtract $3$ from both sides of the equation.

$y = 4 x^{2} - 3$

Step 4

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = 4 x^{2} - 3$

Step 5

Verify if $f^{- 1} (x) = 4 x^{2} - 3$ is the inverse of $f (x) = - \frac{1}{2} \cdot \sqrt{x + 3}$ .

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Step 5.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 5.2

Evaluate $f^{- 1} (f (x))$ .

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Step 5.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 5.2.2

Evaluate $f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 {(- \frac{1}{2} \cdot \sqrt{x + 3})}^{2} - 3$

Step 5.2.3

Simplify each term.

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Step 5.2.3.1

Combine $\sqrt{x + 3}$ and $\frac{1}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 {(- \frac{\sqrt{x + 3}}{2})}^{2} - 3$

Step 5.2.3.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.3.2.1

Apply the product rule to $- \frac{\sqrt{x + 3}}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 ({(- 1)}^{2} {(\frac{\sqrt{x + 3}}{2})}^{2}) - 3$

Step 5.2.3.2.2

Apply the product rule to $\frac{\sqrt{x + 3}}{2}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 ({(- 1)}^{2} (\frac{{\sqrt{x + 3}}^{2}}{2^{2}})) - 3$

Step 5.2.3.3

Raise $- 1$ to the power of $2$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (1 (\frac{{\sqrt{x + 3}}^{2}}{2^{2}})) - 3$

Step 5.2.3.4

Multiply $\frac{{\sqrt{x + 3}}^{2}}{2^{2}}$ by $1$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{{\sqrt{x + 3}}^{2}}{2^{2}}) - 3$

Step 5.2.3.5

Rewrite ${\sqrt{x + 3}}^{2}$ as $x + 3$ .

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Step 5.2.3.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x + 3}$ as ${(x + 3)}^{\frac{1}{2}}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{{({(x + 3)}^{\frac{1}{2}})}^{2}}{2^{2}}) - 3$

Step 5.2.3.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{{(x + 3)}^{\frac{1}{2} \cdot 2}}{2^{2}}) - 3$

Step 5.2.3.5.3

Combine $\frac{1}{2}$ and $2$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{{(x + 3)}^{\frac{2}{2}}}{2^{2}}) - 3$

Step 5.2.3.5.4

Cancel the common factor of $2$ .

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Step 5.2.3.5.4.1

Cancel the common factor.

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{{(x + 3)}^{\frac{2}{2}}}{2^{2}}) - 3$

Step 5.2.3.5.4.2

Rewrite the expression.

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{x + 3}{2^{2}}) - 3$

Step 5.2.3.5.5

Simplify.

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{x + 3}{2^{2}}) - 3$

Step 5.2.3.6

Raise $2$ to the power of $2$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{x + 3}{4}) - 3$

Step 5.2.3.7

Cancel the common factor of $4$ .

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Step 5.2.3.7.1

Cancel the common factor.

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = 4 (\frac{x + 3}{4}) - 3$

Step 5.2.3.7.2

Rewrite the expression.

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = x + 3 - 3$

Step 5.2.4

Combine the opposite terms in $x + 3 - 3$ .

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Step 5.2.4.1

Subtract $3$ from $3$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = x + 0$

Step 5.2.4.2

Add $x$ and $0$ .

$f^{- 1} (- \frac{1}{2} \cdot \sqrt{x + 3}) = x$

Step 5.3

Evaluate $f (f^{- 1} (x))$ .

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Step 5.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 5.3.2

Evaluate $f (4 x^{2} - 3)$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (4 x^{2} - 3) = - \frac{1}{2} \cdot \sqrt{(4 x^{2} - 3) + 3}$

Step 5.3.3

Simplify by adding numbers.

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Step 5.3.3.1

Add $- 3$ and $3$ .

$f (4 x^{2} - 3) = - \frac{1}{2} \cdot \sqrt{4 x^{2} + 0}$

Step 5.3.3.2

Add $4 x^{2}$ and $0$ .

$f (4 x^{2} - 3) = - \frac{1}{2} \cdot \sqrt{4 x^{2}}$

Step 5.3.4

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$f (4 x^{2} - 3) = - \frac{1}{2} \cdot \sqrt{{(2 x)}^{2}}$

Step 5.3.5

Pull terms out from under the radical, assuming positive real numbers.

$f (4 x^{2} - 3) = - \frac{1}{2} \cdot (2 x)$

Step 5.3.6

Cancel the common factor of $2$ .

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Step 5.3.6.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$f (4 x^{2} - 3) = \frac{- 1}{2} \cdot (2 x)$

Step 5.3.6.2

Factor $2$ out of $2 x$ .

$f (4 x^{2} - 3) = \frac{- 1}{2} \cdot (2 (x))$

Step 5.3.6.3

Cancel the common factor.

$f (4 x^{2} - 3) = \frac{- 1}{2} \cdot (2 x)$

Step 5.3.6.4

Rewrite the expression.

$f (4 x^{2} - 3) = - 1 \cdot x$

Step 5.3.7

Rewrite $- 1 x$ as $- x$ .

$f (4 x^{2} - 3) = - x$

Step 5.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = 4 x^{2} - 3$ is the inverse of $f (x) = - \frac{1}{2} \cdot \sqrt{x + 3}$ .

$f^{- 1} (x) = 4 x^{2} - 3$